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All information in these pages is copyright (c) 1989-2003 by Roger Nichols. All rights reserved. Permission for personal reference only, and may not be reproduced by any method without written permission.

Let's Resolve This
by Roger Nichols

We are going to talk about resolution, dither, noise floor, and cumulative errors in digital audio processing. These are the questions I get most on the EQ on-line forum. The subject has to do with 24-bit audio and whether 32 bit or 48 bit internal processing helps, and how much DSP errors affect the audio signal.

Let us first assume that if you are recording 24bit and all of the internal processing is only at 24 bits. Whenever you do anything to the signal, such as change level, or limit, or EQ, or anything, you add about 1.25dB of noise to the signal down at the level of the smallest bit. This is assuming that the process adds no noise of its own because of the process. The math remainders cause the errors due to multiplications and divisions. We are dealing with binary math because digital audio is binary. You cannot have a fraction of a bit (well you can, by using Pulse Width Modulation on the Least Significant Bit, but that is another column.)
Now let's talk about noise floor. This is actually a misnomer in the digital world. This is more of a sensitivity limit. With 24 bits nothing below -144 will be detected or recorded. Noise floor typically means the noise level when no signal is present. Based on this, the noise floor will be the noise of the resistor on the output, because there will be nothing... no signal present at all if all of the bits are zero. If you get your input level up above -144 you will start to record some information. With no noise shaping or dithering your recording will be very distorted, but you can easily tell what is being recorded. At that level you are basically recording with a one-bit converter. As a point of comparison, those announcements you hear at the airport "This area is for loading and unloading of passengers only. All other vehicles will be a toad at the owners expense." The devices that play this announcement are one bit digital recorders. Not bad.

Everyone focuses on this low noise floor and dithering to improve the quality of the signal at low levels, but that is not all there is to it. When you zoom way in to look at a waveform, you see a single line running up and down as it crosses the screen. Change the vertical and horizontal magnification until you can see small vertical moves in the waveform. These are the high frequency overtones of the instrument. Now lower the vertical resolution one step at a time until the little wiggle goes away.
What you have done is changed the magnification until the screen can no longer display the small movement of the waveform because of the size of the pixels on the monitor. This same thing happens when you don’t have enough vertical resolution in an A/D converter. The little wiggle is there, but the size of the LSB is too small to capture it. This happens to the little wiggles in the loud parts too. The waveform could be all the way up to clipping levels, but the small bits are still the ones that determine the resolution of the capture of the little wiggles in the waveform.

Dithering adds about 1.5 bits to the apparent resolution of the recording. With dithering you could detect and record a signal that was 9dB lower than the -144dB "Noise Floor."

Data Transfer
There has been some misunderstanding of how 16-bit or 20-bit data is converted to 24-bit data when importing data from a lower resolution recorder. All digital devices start numbering the bits at the top, or Most Significant Bit. The MSB in PCM (Pulse Code Modulation) audio is the Sign bit. This bit determines whether the sample is negative (below the zero line) or positive. The next 15 bits (for 16-bit audio) set the level of the sample. Audio coding is binary, but let’s just say that the level of this sample is 32760 on a scale of 1 to 32768. Just as the 3 in our number has the most worth (30,000), then the 2 (2,000) then the 7 (700), the 6 (60) and the 8 (8), the bits of our audio sample line up the same way, each one having more worth than the one to its right. In decimal math each digit is worth 10 times the one on its right. In binary arithmetic each bit is worth 2 times the bit to its right.

All PCM digital audio word lengths start with the same MSB, the sign bit. The MSB in a 24-bit word is worth exactly the same. Think of it like a 5-bit cash drawer and a 7-bit cash drawer. The 5bit drawer has a place for $10,000 bills, $1,000 bills, $100 bills, $10 bills and $1 bills. Our 7-bit drawer has the same setup plus a bin for dimes and a bin for pennies.

If we move $11,111 from the 5-bit drawer to the 7-bit drawer there is a place for every denomination. The dime bin and the penny bin remain empty. If we move $11,111.11 from the 7-bit drawer to the 5-bit drawer we have a bin for everything except the dime and penny. Since there is no place for them we must throw them away.

The exact same thing happens when we go back and forth between 16bit and 24bit machines. If we go from 16-bits to 24-bits there is a bin for each bit and the bottom 8bits remain empty. If we go from 24-bits to 16-bits the top 16-bits fit just fine, but the bottom 8 just get thrown away because there is no place to put them. They are truncated.

To carry this further, moving from 16-bit, 20-bit or 24-bit into a 48-bit digital mixer works the same way. All of the extra bins are left empty (set to zero) when data comes in. As soon as we do anything to the data, like change the level, then the extra bits are filled with the results of the math. When it is time to go back to 24-bits, then the extra bits are truncated off and thrown away. This is where dithering comes in.

Dithering For Fun And Profit.Dithering does not add any bits to the 24-bit (or 20-bit or 16-bit) audio signal. It adds low noise to the signal so that the LSB can detect a signal that is below its threshold. I won’t go into the details now, but suffice to say that a dithered 1bit signal sounds better than a 2-bit converter but not as good as a 3-bit converter. So we have added some perceptual enhancement to the signal instead of just chopping it off.

Internal Resolution
OK, now lets talk about 32-bit or 48-bit or floating point or what’s-the-point. The following discussion will include 48-bit internal word length.The errors caused by the math only apply to the smallest bits. If we do the math now, the remainder from the division of a 24-bit number will be below the 24-bit level, somewhere between bit 25 and 48. This means that there will be no round off errors in our 24-bit signal. We can DSP or change levels all we want. There will only be one penalty at the end when the signal is changed back to 24bits. If there is no dither or noise shaping, then the maximum penalty will be 1.25dB at the -144db level.
Noise floors do not add up digitally like they do in analog, but the level can rise. If you add a full level digital signal to another full level digital signal, then the results will be one more bit than you have room for. This will put you 6dB over the clipping limit. Oops. You do not have to worry about the digital noise floor getting high by adding 32 or 48 tracks of stuff. You can mix together 1000 tracks with nothing on them and have -144dB noise level on the output. Try that with an analog console or tape machine.
Digital mixing consoles must accommodate the possibility that all of the signals entering the console may be full level signals. The 32-bit or 48-bit internal busses are usually split up so that some of the bits are used for extra room when you add the signals together, and some of the bits are used for low-level math errors.

Now, what you do have to worry about is the noise from each source. The noise from the synth and the room noise on the vocal and the mic hiss on the horns... All of these noises are 100 times louder than the "noise floor" of the 24-bit converter.

Quantization Error
Earlier we touched on the fact that the converter could not detect anything below the level of the LSB. Actually the level of concern is about half the level of the LSB.

Have you ever noticed the way that your air conditioner thermostat works? The temperature is set at 75 degrees. When the temperature gets up to 76 degrees the central air comes on and cools off the room until the temperature gets to 74 degrees and then shuts off. The central air stays off until the temperature gets back up to 76 degrees and the cycle repeats. This is called hysteresis.
The LSB in the converter works the same way. The level of the audio gets up to about 3/4 of the value of the bit and the bit turns on. Even though the analog signal level is falling from this level, at every sample time (44,100 times per second) the LSB is turned on until the level of the signal reaches about 1/4 of the bit value. When you play back the digital audio signal, the LSB remains on, even though the signal level was falling. After a few samples the bit turns off and stays off. This is not exactly a reciprocal of the input signal. This is quantization error. If you use 24 bits instead of 16 bits the quantization error is 256 times less.

Those of you not interested in this topic please consider it my April fools joke. The rest of you sharpen up your pencils for next month’s column on the correlation between digital audio and fatal gardening accidents.